Evidence of a short-range incommensurate $d$-wave charge order from a fermionic two-loop renormalization group calculation of a 2D model with hot spots

Abstract: The two-loop renormalization group (RG) calculation is considerably extended here for the two-dimensional (2D) fermionic effective field theory model, which includes only the so-called "hot spots" that are connected by the spin-density-wave (SDW) ordering wavevector on a Fermi surface generated by the 2D $t-t'$ Hubbard model at low hole doping. We compute the Callan-Symanzik RG equation up to two loops describing the flow of the single-particle Green's function, the corresponding spectral function, the Fermi velocity, and some of the most important order-parameter susceptibilities in the model at lower energies. As a result, we establish that -- in addition to clearly dominant SDW correlations -- an approximate (pseudospin) symmetry relating a short-range \emph{incommensurate} $d$-wave charge order to the $d$-wave superconducting order indeed emerges at lower energy scales, which is in agreement with recent works available in the literature addressing the 2D spin-fermion model. We derive implications of this possible electronic phase in the ongoing attempt to describe the phenomenology of the pseudogap regime in underdoped cuprates.